| L(s) = 1 | − 3-s + 2·7-s − 2·9-s + 13-s − 6·17-s − 2·19-s − 2·21-s − 23-s − 5·25-s + 5·27-s + 3·29-s + 5·31-s − 8·37-s − 39-s + 3·41-s − 8·43-s + 9·47-s − 3·49-s + 6·51-s − 6·53-s + 2·57-s + 12·59-s − 14·61-s − 4·63-s − 8·67-s + 69-s − 15·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s − 2/3·9-s + 0.277·13-s − 1.45·17-s − 0.458·19-s − 0.436·21-s − 0.208·23-s − 25-s + 0.962·27-s + 0.557·29-s + 0.898·31-s − 1.31·37-s − 0.160·39-s + 0.468·41-s − 1.21·43-s + 1.31·47-s − 3/7·49-s + 0.840·51-s − 0.824·53-s + 0.264·57-s + 1.56·59-s − 1.79·61-s − 0.503·63-s − 0.977·67-s + 0.120·69-s − 1.78·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.808684280597188673489137150163, −8.521126016011552017519779837642, −7.48540449454009034199503917488, −6.49198643404353442573205248031, −5.87052871719786078173316926452, −4.88913521426515505557620586765, −4.19829185799095109044352510087, −2.83296492787843542508096840013, −1.68213519834932597828765870338, 0,
1.68213519834932597828765870338, 2.83296492787843542508096840013, 4.19829185799095109044352510087, 4.88913521426515505557620586765, 5.87052871719786078173316926452, 6.49198643404353442573205248031, 7.48540449454009034199503917488, 8.521126016011552017519779837642, 8.808684280597188673489137150163
